Seminars in Geometry/Topology

Topology Seminar Wednesday 14 February, 13:15-14:00, Gamle Fysikk F4

Alexander Schmeding (NTNU): Splitting sequences of diffeomorphism groups on manifolds with boundary

Abstract: Due to a result by Thurston, the (unit component of the) diffeomorphism group of a compact manifold without boundary is simple. If the compact manifold M has non-empty boundary B, this is no longer true, as the subgroup Diff_1 (M) of all diffeomorphisms restricting to the identity on the boundary is a normal subgroup. Indeed, this gives rise to an interesting short exact sequence of infinite-dimensional Lie groups (where we suppress taking the unit component in the notation):

Diff_1 (M) → Diff (M) → Diff (B)

In this talk I will explain that this sequence of groups actually splits, i.e. there is a smooth section near the identity of the projection map onto Diff (B). Hence one can regard the whole sequence as an infinite-dimensional fibre-bundle. This information seems to be new and we shall sketch a few ideas for applications of this construction.

Topology Seminar Wednesday 28 February, 13:00-14:00, Kjelhuset KJL24 (note the time and room)

Markus Szymik (Sheffield): Buildings for the Cremona groups

Abstract: The Cremona groups are the groups of the birational self-equivalences of the affine and the projective spaces. Despite being basic in birational geometry, these groups are still only poorly understood, and little was known beyond the cases of the projective line and the projective plane until recently. In this talk, I will describe a solution to the problem of finding spaces on which the Cremona groups act nicely and explain how these spaces can be used to show that we can expect to observe general phenomena in large dimensions.

Topology Seminar Wednesday 6 March, 13:00-14:00, Gamle Fysikk F4

Eiolf Kaspersen (NTNU): Geometric constructions for the complex cobordism of Lie groups

Abstract: When examining a space such as a Lie group, it is a classical question which elements of its cohomology can be represented by smooth manifolds. More specifically, we may ask which elements are in the image of the Thom morphism from complex cobordism to singular cohomology. To answer this question, we combine algebraic methods such as cohomology operations with more geometric methods that deal with the group structure of the Lie groups.

In this talk I will present some methods we can use to show that the Thom morphism is not surjective for certain Lie groups. Furthermore, I will give geometric constructions of their complex cobordism to see what exactly it is about the geometry of these groups that makes this phenomenon occur.

Topology Seminar Wednesday 17 April, 13:15-14:00, Gamle Fysikk F4

Morten Brun (Bergen): Density dependent topological data analysis

Abstract: I will review approaches to density dependent topological data analysis given by the multicover bifiltration and distance to measure, and formulate them as part of a common generalization. I will also demonstrate how these filtrations can be used to treat big amounts of data by computing persistent homology of clusters.

Topology Seminar Wednesday 24 April, 13:15-14:15, Kjelhuset KJL24 (note the room)

William Hornslien (NTNU): Combing a hedgehog over a field

Abstract: The hedgehog theorem (also known as the hairy ball theorem), is one of the most well-known results in topology. The theorem states that for that vector fields over the n-sphere must vanish somewhere when positive and even. The sphere is defined by the equation x_1^2 + … + x_{n+1}^2=1, and the classical result is when this is viewed as an equation over the reals. Can one say something similar when we consider the equation over any field? Using motivic homotopy theory, we are able to use the same argument as in differential topology, but with some complications and detours along the way. This talk is based on a paper of the same title by Alexey Ananyevskiy and Marc Levine.